1. Field of the Invention
This invention relates generally to the field of fluid dynamics. More particularly, the invention is a method for accurately and efficiently analyzing anisotropic turbulent flows of both gases and liquids.
2. Background
The study and engineering application of hydraulics or fluid mechanics dates to the dawn of civilization. The first irrigation canals were constructed before 5,000 BC. Theoretical analysis of fluid motions began in the modem scientific sense with Newton's Principia (1687) and his laws of motion. Bernoulli (1738) established a relationship between the pressure and velocity of an ideal fluid which became his famous theorem. d'Alembert (1744) deduced that the steady flow of an ideal fluid about a body produces no drag (d'Alembert's Paradox). Euler (1755) founded modem fluid mechanic analyses by deriving the differential equations of motion (continuity and momentum conservation) for an ideal (inviscid) isotropic fluid. Navier (1822) and Stokes (1845) independently derived the viscous terms necessary to extend Euler's equations to isotropic, viscous fluids. The equation of state and the conservation of energy equation (First Law of Thermodynamics) written utilizing Fourier's (1822) law of thermal energy transfer completed the classical Navier Stokes equations governing isotropic fluid motions.
Reynolds (1883) experimentally showed that there are two modes of fluid motion, laminar and turbulent motion, which occurs at large values of the dimensionless Reynolds number. Reynolds (1894), by considering a time average motion and a time dependent turbulent motion, showed that turbulent motions introduce additional turbulent (Reynolds) stresses into the Navier Stokes equations, greatly increasing the number of flow properties required to describe turbulent flows.
Despite over a century of intense aircraft and aerospace developments, wind tunnel and laboratory experiments, theoretical analyses and numerical computations based on the Navier Stokes equations, the determination of a general theory of turbulent motion remains one of the last great unsolved problems of classical physics.
Maxwell (1858) founded modern kinetic theory when he introduced his new concept of the molecule velocity distribution function and deduced its equilibrium form. Maxwell (1867) derived the Maxwell transport equations and showed that the collision change integrals could be solved analytically without knowing the molecule velocity distribution function for Maxwell molecules. Boltzmann (1872) derived his integro-differential equation for Maxwell's molecular velocity distribution function and solved for his famous H-theorem, which proved that Maxwell's equilibrium form was correct. Chapman (1916), using Maxwell's transport equations, determined accurate general formulae for the gas transport coefficients. Enskog (1917) gave a general solution method for the Boltzmann equation, which showed that the Euler equations were the first and the Navier Stokes equations were the second approximate solutions to the Boltzmann equation and gave gas transport coefficients identical to Chapman's. Both the Chapman and Enskog analyses were for isotropic perfect gases, being based on perturbations of Maxwell's isotropic equilibrium solution. Their independent analyses cemented the belief in both the scientific and particularly the engineering professions that the Navier Stokes equations were generally and universally valid and that the correct way of analyzing both laminar and turbulent flows was through the Navier Stokes equations. This belief has been engraved in stone in both the scientific and engineering literature throughout the 20th century.
Burnett (1935) derived the third approximate solution to the Boltzmann equations using Enskog's solution method. Grad (1949) derived a thirteen moment method of solving Boltzmann's equation which closely followed Enskog's methods. Both of these analyses have been proven to lack generality and have not been significant advances of the Navier Stokes theory. Bird (1963) introduced the Direct Simulation Monte Carlo (DSMC) numerical method for calculating rarified gas flows. Yen (1966) showed that the directional thermal energies (temperatures) were vastly different in shock waves and that the longitudinal temperature overshot its downstream value for shock Mach numbers greater than
            9      5        ⁢            (      1.34      )        .  Elliot and Baganoff (1974) showed that the Navier Stokes normal stress relationship and the Fourier energy flux component ratio were valid only in sound waves and were invalid at the shock end points. Elliot (1975) showed that the Navier Stokes normal stress relationship was invalid everywhere in shock waves and incompatible with the directional thermal kinetic energy moments of the Boltzmann equation. DSMC numerical shock calculations confirmed all of these analytical predictions and numerically illustrated the anisotropic not isotropic fluid features of many simple gas flows.
Kliegel (1990), following a suggestion of Maxwell (1867) that near equilibrium flows were anisotropic and should be represented by an anisotropic, not isotropic Maxwellian, performed the first general anisotropic solution analysis of the Boltzmann equation. Kliegel (1990) showed that the Euler-Navier Stokes-Burnett equation sequence was not the correct approximate solution sequence for the Boltzmann equation. He showed that the correct gas dynamic equation set was an anisotropic fluid seven equation set for the density, three fluid velocity components and three directional thermal kinetic energies (temperatures), not the classic isotropic fluid five equation Euler-Navier Stokes set for the density, three fluid velocity components and total temperature. He also gave shear and directional energy flux relationship having the correct Mach number dependence for both sound waves and weak shocks. Kliegel's anisotropic equations resolved d'Alembert's paradox giving a profile pressure drag associated with pushing an ideal (shearless, energy fluxless) fluid about a body. They also correctly predicted the directional thermal energy separations and overshoots and the thermal energy flux component changes occurring in shock waves as predicted by Yen (1966) and numerical DSMC shock calculations. Bird (1994) summarizes recent DSMC calculational capabilities. Chen and Jaw (1998) presented a recent summary of classical isotropic fluid Navier Stokes based turbulent flow modelling.
The present disclosure correctly consolidates all previous fluid dynamic and kinetic theory analyses and extends it to define a new method of analyzing and computing anisotropic turbulent flows.
The prior art cited above may be found in the following references:    1. 1687, Newton, I. Philosophiae Naturalis Principia Mathematica, Oxford.    2. 1738, Bernoulli, D., Hydrodynamica sive de Viribus et Motibus Fluidorum Commentarii, Berlin.    3. 1744, d'Alembert, J., Traite de l'equilibre et du mouvement des fluides pour servir de suite au traité de dynamique, Paris.    4. 1755, Euler, L., Principles généreaux du mouvement des fluides, Histoire de l'Academie de Berlin.    5. 1822, Navier, L. M. H., Mémoires de l'Academie des Sciences de l'Institut de France, V 6, p. 389, Paris.    6. 1822, Fourier, J., Théorie analytique de la chaleur, Paris.    7. 1845, Stokes, G. G., On the Theory of the Internal Friction of Fluids in Motion, Transactions of the Cambridge Philosophical Society, V8, p. 287, Cambridge.    8. 1858, Maxwell, J. C., Illustrations of the Dynamical Theory of Gases, Philosophical Magazine, V 19, p. 19 and V20, p. 21.    9. 1867, Maxwell, J. C., On the Dynamical Theory of Gases, Philosophical Transactions of the Royal Society, V 157, p. 49.    10. 1872, Boltzmann, L., Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, Styungsberichte, Akad. Wiss, Vienna, Part II, V 66, p. 275.    11. 1883, Reynolds, O., An Experimental Investigation of the Circumstances which Determine whether the Motion of Water Shall Be Direct or Sinuous, and the Law of Resistance in Parallel Channels, Philosophical Transactions of the Royal Society, V174, p. 935.    12. 1895, Reynolds, O., On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion, Philosophical Transactions of the Royal Society, V186A, p. 123.    13. 1904, Prandth, L., Uber Flüssigkeitsbewegung bei sehr kleiner Reibung, Proceedings of the Third International Mathematical Conference, Heidelberg, Leipzig.    14. 1916, Chapman, S., The Kinetic Theory of Simple and Composite Monotonic Gases: Viscosity, Thermal Conduction, and Diffusion, Proceedings of the Royal Society London, A, V 93, p. 1.    15. 1917, Enskog, D., Kinetische Theory der Vorgänge in mässig verdunnten Gasen, Almquist & Wilsells, Uppsala, Sweden.    16. 1935 Burnett, D., The Distribution of Molecular Velocities and the Mean Motion in a Non-Uniform Gas, Proceedings London Mathematical Society, V 40, p. 382.    17. 1949, Grad, H., On the Kinetic Theory of Rarefied Gases, Convention of Pure and Applied Mathematics, V 2, p. 331.    18. 1963, Bird, G. A., Approach to Translational Equilibrium in a Rigid Sphere Gas, Physics of Fluids, V 6, p. 1518.    19. 1966, Yen, S-M, Temperature Overshoot in Shock Waves, Physics of Fluids, V9, p. 1417.    20. 1970, Chapman, S., Cowling, T. G., Mathematical Theory of Non-Uniform Gases, Third Edition, Cambridge University Press.    21. 1974, Elliot, J. P., Baganoff, D., Solution of the Boltzmann Equation at the Upstream and Downstream Singular Points in a Shock Wave, Journal of Fluid Mechanics, V 65, p. 603.    22. 1975, Elliot, J. P., On the Validity of the Navier-Stokes Relation in a Shock Wave, Canadian Journal of Physics, V 53, p. 583.    23. 1990, Kliegel, J. R., Maxwell Boltzmann Gas Dynamics, Proceedings of the 17th International Symposium on Rarefied Gas Dynamics, Aachen 1990, Edited by Alfred E. Beylich, VCH, New York.    24. 1994, Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, New York.    25. 1998, Chen, C-J. Jaw, S-Y, Fundamentals of Turbulence Modeling, Taylor & Francis, Washington.